Proof of the limit f(x) = 1/x is 1/3 near 3

1. The problem statement, all variables and given/known data
f(x) = 1/x
Proof that the limit of f(x) = 1/3 near 3
the answer provided by the text is: (1/x - 1/3) < e by requiring abs(x-3) < min(6e, 1)
I can see that the text has the right process and therefore it reaches the right answer, but I have done it in a different process which I believe is also correct but to a different answer. Can you see where have I made the mistake in my process?

2. Relevant equations

3. The attempt at a solution
This is my process:
Note:
(1/x - 1/3) = (3-x)/3x